On the existence of fixed points that belong to the zero set of a certain function

(1)

R E S E A R C H

Open Access

On the existence of ﬁxed points that

belong to the zero set of a certain function

Erdal Karapinar

1,2

_{, Donal O’Regan}

2,3

_{and Bessem Samet}

4*

*_{Correspondence:} [email protected] 4_{Department of Mathematics,} College of Science, King Saud University, P.O. Box 2455, Riyadh, 11451, Saudi Arabia

Full list of author information is available at the end of the article

Abstract

LetT:X→Xbe a given operator andFTbe the set of its ﬁxed points. For a certain function

ϕ

:X→[0,∞), we say thatFTis

ϕ-admissible if

FTis nonempty andFT⊆Zϕ, whereZϕis the zero set of

ϕ. In this paper, we study the

ϕ-admissibility of a new class

of operators. As applications, we establish a new homotopy result and we obtain a partial metric version of the Boyd-Wong ﬁxed point theorem.

MSC: 54H25; 47H10

Keywords:

ϕ-admissible; ﬁxed point; homotopy result; partial metric

1 Introduction

Let (X,d) be a metric space. For a given functionϕ:X→[,∞), we deﬁne the set

Zϕ=

x∈X:ϕ(x) = .

LetT :X→Xbe a given operator. The set of ﬁxed points ofT is denoted byFT, that

is,

FT={x∈X:Tx=x}.

Deﬁnition . We say that the setFTisϕ-admissible if and only ifFT=∅andFT⊆Zϕ.

Let F be the set of functions F : [,∞) _→_[,_∞_{) satisfying the following}

condi-tions:

(F) max{a,b} ≤F(a,b,c), for alla,b,c≥; (F) F(a, , ) =a, for alla≥;

(F) Fis continuous.

As examples, the following functions belong toF:

. F(a,b,c) =a+b+c,

. F(a,b,c) =max{a,b}+ln(c+ ), . F(a,b,c) =a+b+c(c+ ), . F(a,b,c) = (a+b)ec_,

. F(a,b,c) = (a+b)(c+ )n_,_n_∈_N_.

(2)

Letbe the set of functionsψ: [,∞)→[,∞) satisfying the following conditions:

() ψ is upper semi-continuous from the right; () ψ(t) <t, for allt> .

For given functionsϕ:X→[,∞),F∈F, andψ∈, we denote byT(ϕ,F,ψ) the class of operatorsT:X→Xsatisfying

Fd(Tx,Ty),ϕ(Tx),ϕ(Ty)≤ψFd(x,y),ϕ(x),ϕ(y), (x,y)∈X×X. (.)

The aim of this paper is to study theϕ-admissibility of the setFT, whereTbelongs to

the class of operatorsT(ϕ,F,ψ), (F,ψ)∈F×. As applications, we obtain an homotopy result and a partial metric version of the Boyd-Wong ﬁxed point theorem.

2 Main result

Our main result is given in the following theorem.

Theorem . Let(X,d)be a complete metric space and T:X→X be a given operator.

Suppose that the following conditions hold:

(i) there existϕ:X→[,∞),F∈F,andψ∈such thatT∈T(ϕ,F,ψ); (ii) ϕis lower semi-continuous.

Then the set FTisϕ-admissible.Moreover,the operator T has a unique ﬁxed point.

Proof Letξ be an arbitrary element of the setFT. Takex=y=ξin (.), and we get

F,ϕ(ξ),ϕ(ξ)≤ψF,ϕ(ξ),ϕ(ξ). (.)

IfF(,ϕ(ξ),ϕ(ξ))= , from (), we get

ψF,ϕ(ξ),ϕ(ξ)<F,ϕ(ξ),ϕ(ξ),

which is impossible from (.). Consequently, we have

F,ϕ(ξ),ϕ(ξ)= .

Using the above equality and (F), we obtain

ϕ(ξ)≤F,ϕ(ξ),ϕ(ξ)= ,

which yields

ϕ(ξ) = .

Consequently, we have

(3)

Now, we have to prove thatFT is a nonempty set. Letxbe an arbitrary element ofX.

Consider the Picard sequence{xn} ⊂Xdeﬁned by

xn=Tnx, n∈N={, , , . . .},

whereTnis thenth iterate ofT. If for someN∈Nwe havexN=xN+, thenxN will be an

element ofFT. As a result we can suppose that

d(xn,xn+) > , n∈N. (.)

Using (.), we have

Fd(Txn,Txn–),ϕ(Txn),ϕ(Txn–)

≤ψFd(xn,xn–),ϕ(xn),ϕ(xn–)

, n∈N∗; (.)

hereN∗={, , . . .}. If for someN∈N∗, we have

Fd(xN,xN–),ϕ(xN),ϕ(xN–)

= ,

then property (F) yields

d(xN,xN–)≤F

d(xN,xN–),ϕ(xN),ϕ(xN–)

= ,

which is a contradiction with (.). Thus

Fd(xn,xn–),ϕ(xn),ϕ(xn–)

> , n∈N∗. (.)

Using (.), (.), the deﬁnition of the sequence{xn}, and (), we have

⎧ ⎨ ⎩

F(d(xn+,xn),ϕ(xn+),ϕ(xn))≤ψ(F(d(xn,xn–),ϕ(xn),ϕ(xn–))),

ψ(F(d(xn,xn–),ϕ(xn),ϕ(xn–))) <F(d(xn,xn–),ϕ(xn),ϕ(xn–)),

n∈N∗. (.)

It follows immediately from (.) that there exists somec≥ such that

lim

n→∞F

d(xn+,xn),ϕ(xn+),ϕ(xn)

= lim

n→∞ψ

=c. (.)

Suppose now thatc> . Using the properties ()-(), we deduce from (.) that

c=lim sup

n→∞

ψFd(xn,xn–),ϕ(xn),ϕ(xn–)

≤ψ(c) <c,

which is a contradiction. As a consequence, we havec= , that is,

lim

n→∞F

d(xn+,xn),ϕ(xn+),ϕ(xn)

= lim

n→∞ψ

(4)

Using (F) and (.), we get

lim

n→∞d(xn+,xn) =nlim→∞ϕ(xn) = . (.)

Next, we show that{xn}is a Cauchy sequence in the metric space (X,d). Suppose that{xn}

is not a Cauchy sequence. Then there existsε>  for which we can ﬁnd two sequences of positive integers{m(k)}and{n(k)}such that, for allk∈N,

n(k) >m(k) >k, d(xm(k),xn(k))≥ε, d(xm(k),xn(k)–) <ε. (.)

Using (.), for allk∈Nwe have

ε≤d(xm(k),xn(k))

≤d(xm(k),xn(k)–) +d(xn(k)–,xn(k))

< ε+d(xn(k)–,xn(k)),

which yields

ε≤d(xm(k),xn(k)) <ε+d(xn(k)–,xn(k)), k∈N. (.)

Lettingk→ ∞in the above inequality and using (.), we obtain

lim

k→∞d(xm(k),xn(k)) =ε

+ _i.e. lim

k→∞d(xm(k),xn(k)) =ε and d(xm(k),xn(k))≥ fork∈N.

(.)

Using the properties (F)-(F), (.), and (.), we get

lim

k→∞F

d(xn(k),xm(k)),ϕ(xn(k)),ϕ(xm(k))

=F(ε, , ) =ε+.

Using the above limit and (), we obtain

lim sup

k→∞ ψ

Fd(xn(k),xm(k)),ϕ(xn(k)),ϕ(xm(k))

≤ψ(ε). (.)

On the other hand, using (.) and (F), for allk∈Nwe have

ε≤d(xn(k),xm(k))≤d(xn(k),xn(k)+) +d(xn(k)+,xm(k)+) +d(xm(k)+,xm(k))

≤d(xn(k),xn(k)+) +F

d(xn(k)+,xm(k)+),ϕ(xn(k)+),ϕ(xm(k)+)

+d(xm(k)+,xm(k))

≤d(xn(k),xn(k)+) +ψ

Fd(xn(k),xm(k)),ϕ(xn(k)),ϕ(xm(k))

+d(xm(k)+,xm(k)).

Passing to the limit superior ask→ ∞, using (.), (.), and (), we obtain

(5)

which is a contradiction. As a consequence,{xn}is a Cauchy sequence. Since (X,d) is a

complete metric space, there is az∈Xsuch that

lim

n→∞d(xn,z) = . (.)

Sinceϕis lower semi-continuous, it follows from (.) and (.) that

≤ϕ(z)≤lim inf

n→∞ ϕ(xn) = ,

which yields

z∈Zϕ. (.)

Now we show thatz∈FT. Using (.), (F), and (.), we have

d(xn+,Tz)≤ψ

Fd(xn,z),ϕ(xn), 

, n∈N. (.)

Also using the continuity ofF, (F), (.), and (.), we have

lim

n→∞F

d(xn,z),ϕ(xn), 

=F(, , ) = .

Note that from (), we have

lim

t→+ψ(t) = . Then

lim

n→∞ψ

Fd(xn,z),ϕ(xn), 

= lim

t→+ψ(t) = . (.)

Now, passingn→ ∞in (.) and using (.), we get

lim

n→∞d(xn+,Tz) = .

The uniqueness of the limit yields z =Tz. Thus FT is a nonempty set, and the ϕ

-admissibility ofFTis proved. Finally, in order to prove the uniqueness of the ﬁxed point, let

us assume thatw∈FTwithd(z,w) > . SinceFTisϕ-admissible, we know thatz,w∈Zϕ.

Now, applying (.) with (x,y) = (z,w), we obtain

Fd(z,w), , ≤ψFd(z,w), , .

Using the properties (F) and (), we get

d(z,w)≤ψd(z,w)<d(z,w),

(6)

Remark . Takeϕ≡ andF(a,b,c) =a+b+cin Theorem ., and we recover the Boyd-Wong ﬁxed point theorem [].

Now, we give some examples to illustrate our main result given by Theorem ..

Example . We endow the setX= [,∞) with the standard metric

d(x,y) =|x–y|, (x,y)∈X×X. LetT:X→Xbe the mapping deﬁned by

Tx=

⎧ ⎨ ⎩

 if ≤x≤,

x

 if  <x.

Observe thatTis not continuous inX. So, there is noψ∈such that

d(Tx,Ty)≤ψd(x,y), (x,y)∈X×X.

Then the Boyd-Wong ﬁxed point theorem cannot be applied in this case. Let ϕ:X→

[,∞) be the function deﬁned by

ϕ(x) =xn, x∈X, for somen∈N∗.

LetF: [,∞)_→_[,_∞_{) be the function deﬁned by}

F(a,b,c) =a+b+c, a,b,c≥.

Letψ: [,∞)→[,∞) be the function deﬁned by

ψ(t) = t

, t≥.

Observe that F belongs to the set F andψ belongs to the set . We claim that T ∈

T(ϕ,F,ψ), that is,

d(Tx,Ty) +ϕ(Tx) +ϕ(Ty)≤ 

 d(x,y) +ϕ(x) +ϕ(y)

, (x,y)∈X×X. (.) In order to prove our claim, we distinguish three cases.

Case. (x,y)∈[, ]×[, ]. In this case, we have

d(Tx,Ty) +ϕ(Tx) +ϕ(Ty) = ≤ 

 d(x,y) +ϕ(x) +ϕ(y)

.

Case. (x,y)∈[, ]×(,∞). In this case, we have

d(Tx,Ty) +ϕ(Tx) +ϕ(Ty) =y +

y



n

(7)

while



 d(x,y) +ϕ(x) +ϕ(y)

=

 y–x+x

n₊_yn_.

Then we have to prove that

yn

 n–– 

≤xn–x.

Observe that the functionh: [, ]→Rdeﬁned by

h(x) =xn–x, x∈[, ],

has a global minimum atxn= (_n)



n– _{which is equal to}_x_n₍–n

n )≥

–n

n . So, we have just to

check that

yn

 n–– 

≤

n– .

Sincey> , we have

yn

 n–– 

≤ 

n–– ≤



n– .

Then our claim holds in this case.

Case.x,y> . In this case, we have

d(Tx,Ty) +ϕ(Tx) +ϕ(Ty) =|x–y|

 +

x



n

+

y



n

and



 d(x,y) +ϕ(x) +ϕ(y)

=|x–y|

 +

xn+yn

 .

Obviously, we have

 d(x,y) +ϕ(x) +ϕ(y)

.

Finally, in all cases our claim (.) holds, which yieldsT∈T(ϕ,F,ψ). By Theorem ., the setFT isϕ-admissible andT has a unique ﬁxed point. In this example,FT={}and

ϕ() = .

Example . We endow the setX= [√,∞) with the standard metric

(8)

LetT:X→Xbe the mapping deﬁned by

Tx=

⎧ ⎨ ⎩

√

 if√≤x≤√,

x

 if 

√  <x.

As in the previous example, the Boyd-Wong ﬁxed point theorem cannot be applied in this case. Letϕ:X→[,∞) be the function deﬁned by

ϕ(x) =x– , x∈X.

LetF: [,∞)_→_[,_∞_{) be the function deﬁned by}

Letψ: [,∞)→[,∞) be the function deﬁned by

ψ(t) = t

, t≥. We distinguish three cases.

Case. (x,y)∈[√, √]×[√, √]. In this case, we have

d(Tx,Ty) +ϕ(Tx) +ϕ(Ty) = ≤ 

 d(x,y) +ϕ(x) +ϕ(y)

.

Case. (x,y)∈[√, √]×(√,∞). In this case, we have

d(Tx,Ty) +ϕ(Tx) +ϕ(Ty) =y –

√  +y



 – ,

while



 d(x,y) +ϕ(x) +ϕ(y)

=y –

x

+

x

 +

y

 – .

Clearly, we have

 d(x,y) +ϕ(x) +ϕ(y)

.

Case. (x,y)∈(√,∞). In this case, we have

d(Tx,Ty) +ϕ(Tx) +ϕ(Ty) =|x–y|

 +

x

 +

y

 – ,

while



 d(x,y) +ϕ(x) +ϕ(y)

=|x–y|

 +

x

 +

y

(9)

Also, we have

 d(x,y) +ϕ(x) +ϕ(y)

.

As a consequence, the mappingT belongs toT(ϕ,F,ψ). By Theorem ., the setFT is

ϕ-admissible andThas a unique ﬁxed point. In this example,FT={

√

}andϕ(√) = .

Example . Let (X,d) be the metric space considered in Example .. We take the func-tionsϕ,F, andψ deﬁned in Example .. LetT:X→Xbe the mapping deﬁned by

Tx=

⎧ ⎨ ⎩

√

 if√≤x≤√, sinx

 if 

√  <x.

Similarly, we haveT∈T(ϕ,F,ψ). By Theorem ., the setFTisϕ-admissible andThas a

unique ﬁxed point. In this example,FT={

√

}andϕ(√) = .

3 Applications

3.1 An homotopy result

Let us denote byF∗the set of functionsF∈Fsatisfying the following property:

(F) for alla,b,c,d≥,

a≤c+d ⇒ F(a,b, )≤F(c,b, ) +d. As examples, the following functions belong toF∗:

. F(a,b,c) = (a+b)ec_,

. F(a,b,c) = (a+b)(c+ )n_,_n_∈_N_.

Observe thatF∗F. To see this, let us consider the function

F(a,b,c) =aec+b+bea+c, a,b,c≥. It is not diﬃcult to check thatF∈FbutF∈/F∗.

We have the following homotopy result.

Theorem . Let(X,d)be a complete metric space, U be an open subset of X,and V be a closed subset of X with U ⊂V.Suppose that H:V ×[, ]→X has the following properties:

(C) x=H(x,λ)for everyx∈V\Uandλ∈[, ];

(C) there exist a continuous functionϕ:X→[,∞),L∈(, ),andF∈F∗such that for allx,y∈Vandλ∈[, ],

FdH(x,λ),H(y,λ),ϕH(x,λ),ϕH(y,λ)≤LFd(x,y),ϕ(x),ϕ(y);

(C) there exists a continuous functionη: [, ]→Rsuch that for allx∈Vandλ,μ∈[, ],

FdH(x,λ),H(x,μ),ϕH(x,λ),ϕH(x,μ)≤η(λ) –η(μ).

(10)

Proof Suppose thatH(·, ) has a ﬁxed point. Consider the set

Q=t∈[, ] :x=H(x,t) for somex∈U.

From (C), clearly  is an element of Q, soQis a nonempty set. We will show that Q

is both closed and open in [, ], and so by the connectedness of [, ], we are ﬁnished sinceQ= [, ]. First, let us prove that Qis open in [, ]. Lett∈Qandx∈U with

x=H(x,t). Using (C) withx=y=xandλ=t, we obtain

F,ϕ(x),ϕ(x)

≤LF,ϕ(x),ϕ(x)

,

which implies sinceL∈(, ) that

F,ϕ(x),ϕ(x)

= .

Then (F) yields

ϕ(x) = .

Moreover, observe that, for allt∈[, ], ifx∈Uis a ﬁxed point ofH(·,t), thenϕ(x) = . On the other hand, sinceUis open in (X,d), there existsr>  such thatB(x,r)⊆U, where

B(x,r) =

z∈X:d(x,z) <r

.

Consider the set

(x,ϕ) =

z∈X:Fd(z,x),ϕ(z), 

<r.

Clearly (x,ϕ) is nonempty (since x ∈ (x,ϕ)) and (x,ϕ)⊆ B(x,r). Let ε =

( –L)r> . Sinceηis continuous ont, there existsα(ε) >  such that

t∈t–α(ε),t+α(ε)

∩[, ] ⇒ η(t) –η(t)<ε.

Lett∈(t–α(ε),t+α(ε))∩[, ]. Forx∈(x,ϕ) (the closure of(x,ϕ)), we have

FdH(x,t),x

,ϕH(x,t), =FdH(x,t),H(x,t)

,ϕH(x,t), . Also since

dH(x,t),H(x,t)

≤dH(x,t),H(x,t)

+dH(x,t),H(x,t)

,

using the properties (F), (F) we get

FdH(x,t),H(x,t)

,ϕH(x,t),  ≤FdH(x,t),H(x,t)

,ϕH(x,t), +dH(x,t),H(x,t)

≤FdH(x,t),H(x,t)

,ϕH(x,t), +FdH(x,t),H(x,t)

,ϕH(x,t)

(11)

≤η(t) –η(t)+LF

d(x,x),ϕ(x), 

<ε+Lr=r.

Thus we proved that, for allt∈(t–α(ε),t+α(ε))∩[, ], the operator

H(·,t) :(x,ϕ)→(x,ϕ)

is well deﬁned. Now, using (C) and Theorem ., we deduce that, for allt∈(t–α(ε),

t+α(ε))∩[, ], the operatorH(·,t) has a ﬁxed point inV. However, such a ﬁxed point

should be inUfrom (C). As a consequence,

t–α(ε),t+α(ε)

∩[, ]⊆Q,

which proves thatQis open in [, ]. Next, we show thatQis closed in [, ]. To see this, let{tn}be a sequence inQwithtn→t∈[, ] asn→ ∞. We have to prove thatt∈Q.

From the deﬁnition ofQ, for alln∈N, there existsxn∈Uwith

xn=H(xn,tn) and ϕ(xn) = .

Also for allm,n∈N, we have

d(xn,xm) =d

H(xn,tn),H(xm,tm)

≤dH(xn,tn),H(xn,tm)

+dH(xn,tm),H(xm,tm)

≤FdH(xn,tn),H(xn,tm)

,ϕH(xn,tn)

,ϕH(xn,tm)

+FdH(xn,tm),H(xm,tm)

,ϕH(xn,tm)

,ϕH(xm,tm)

≤η(tn) –η(tm)+LF

d(xn,xm), , 

=η(tn) –η(tm)+Ld(xn,xm),

which yields

d(xn,xm)≤|

η(tn) –η(tm)|

 –L , m,n∈N.

Lettingm,n→ ∞in the above inequality and using the continuity ofη, we getd(xn,xm)→

 asm,n→ ∞, which implies that{xn}is a Cauchy sequence in the complete metric space

(X,d). Then there is somez∈V(sinceV is closed) such that

lim

n→∞d(xn,z) =  and ϕ(z) = ,

sinceϕis lower semi-continuous. Now, for alln∈Nwe have

dxn,H(z,t)

=dH(xn,tn),H(z,t)

≤dH(xn,tn),H(xn,t)

+dH(xn,t),H(z,t)

≤FdH(xn,tn),H(xn,t)

,ϕH(xn,tn)

,ϕH(xn,t)

+FdH(xn,t),H(z,t)

,ϕH(xn,t)

(12)

≤η(tn) –η(t)+LF

d(xn,z), , 

=η(tn) –η(t)+Ld(xn,z).

Lettingn→ ∞in the above inequality, we obtain

lim

n→∞d

xn,H(z,t)

= .

The uniqueness of the limit yieldsz=H(z,t). Using (C), we deduce thatz∈Uandt∈Q. ThusQis closed in [, ].

For the reverse implication, we use the same technique.

3.2 A partial metric version of Boyd-Wong ﬁxed point theorem

In this part, using Theorem ., we obtain a partial metric version of the Boyd-Wong ﬁxed point theorem.

We start by recalling some basic deﬁnitions and properties of partial metric spaces. For more details of such spaces, we refer the reader to [–].

A partial metric on a nonempty setXis a functionp:X→X→[,∞) such that for all

x,y,z∈X, we have

(i) p(x,x) =p(y,y) =p(x,y)⇐⇒x=y; (ii) p(x,x)≤p(x,y);

(iii) p(x,y) =p(y,x);

(iv) p(x,y)≤p(x,z) +p(z,y) –p(z,z).

A partial metric space is a pair (X,p) such thatXis a nonempty set andpis a partial metric onX. It is clear that, ifp(x,y) = , then from (i)-(ii),x=y; but ifx=y,p(x,y) may not be . A basic example of a partial metric space is the pair ([,∞),p), wherep(x,y) =max{x,y}.

Each partial metric ponXgenerates a Ttopologyτp onX which has as a base the

family of openp-balls{Bp(x,ε) :x∈X,ε> }, where

Bp(x,ε) :=

y∈X:p(x,y) <p(x,x) +ε.

Let (X,p) be a partial metric space. A sequence{xn} ⊂Xconverges to somex∈Xwith

respect topif and only if

lim

n→∞p(xn,x) =p(x,x).

A sequence{xn} ⊂X is said to be a Cauchy sequence if and only iflimm,n→∞p(xn,xm)

exists and is ﬁnite. The partial metric space (X,p) is said to be complete if and only if every Cauchy sequence{xn}inXconverges to somex∈Xsuch thatlimn,m→∞p(xn,xm) =p(x,x).

Ifpis a partial metric onX, then the functiondp:X→X→[,∞) deﬁned by

dp(x,y) = p(x,y) –p(x,x) –p(y,y), (x,y)∈X, (.)

is a metric onX.

Lemma . Let(X,p)be a partial metric space.Then:

(i) {xn}is a Cauchy sequence in(X,p)if and only if{xn}is a Cauchy sequence in the

(13)

(ii) the partial metric space(X,p)is complete if and only if the metric space(X,dp)is

complete.Furthermore,limn→∞dp(xn,x) = if and only if

lim

n→∞p(xn,x) =p(x,x) =m,limn→∞p(xn,xm).

We have the following result.

Corollary . Let(X,p)be a complete partial metric space and let T:X→X be an oper-ator such that

p(Tx,Ty)≤ψp(x,y), (x,y)∈X×X, (.)

whereψ∈.We have the following results:

(i) ifz∈Xis a ﬁxed point ofTthenp(z,z) = ; (ii) T has a unique ﬁxed point.

Proof Letdpbe the metric onXdeﬁned by (.). We have

p(x,y) =d(x,y) +ϕ(x) +ϕ(y), (x,y)∈X×X,

where

d(x,y) = dp(x,y)

 , ϕ(x) =

p(x,x)  .

Then (.) yields

Fd(Tx,Ty),ϕ(Tx),ϕ(Ty)≤ψFd(x,y),ϕ(x),ϕ(y), (x,y)∈X×X,

where

From (ii) Lemma ., the metric space (X,d) is complete and the functionϕis continuous with respect to the topology ofd. Finally the desired result follows from Theorem ..

Remark . Take in Corollary .,ψ(t) =ktwith k∈(, ), and we recover Matthews ﬁxed point theorem [].

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally and signiﬁcantly in writing this article. All authors read and approved the ﬁnal manuscript.

Author details

1_{Department of Mathematics, Atilim University, Incek, Ankara, 06836, Turkey.}2_{Nonlinear Analysis and Applied}

(14)

Acknowledgements

The third author would like to extend his sincere appreciation to the Deanship of Scientiﬁc Research at King Saud University for its funding of this research through the International Research Group Project No. IRG14-04.

Received: 16 May 2015 Accepted: 10 August 2015

References

1. Boyd, DW, Wong, JSW: On nonlinear contractions. Proc. Am. Math. Soc.20, 458-464 (1969)

2. Abdeljawad, T, Karapinar, E, Ta¸s, K: A generalized contraction principle with control functions on partial metric spaces. Comput. Math. Appl.63(3), 716-719 (2012)

3. Altun, I, Acar, O: Fixed point theorems for weak contractions in the sense of Berinde on partial metric spaces. Topol. Appl.159, 2642-2648 (2012)

4. Argarwal, RP, Alghamdi, MA, Shahzad, N: Fixed point theory for cyclic generalized contractions in partial metric spaces. Fixed Point Theory Appl.2012, 40 (2012)

5. Ciric, L, Samet, B, Aydi, H, Vetro, C: Common ﬁxed points of generalized contractions on partial metric spaces and an application. Appl. Math. Comput.218, 2398-2406 (2011)

6. Haghi, RH, Rezapour, S, Shahzad, N: Be careful on partial metric ﬁxed point results. Topol. Appl.160(3), 450-454 (2013) 7. Huang, XJ, Li, YY, Zhu, CX: Multivaluedf-weakly Picard mappings on partial metric spaces. J. Nonlinear Sci. Appl.8,

1234-1244 (2015)

8. Jleli, M, Karapinar, E, Samet, B: Further remarks on ﬁxed point theorems in the context of partial metric spaces. Abstr. Appl. Anal.2013, Article ID 715456 (2013)

9. Matthews, SG: Partial metric topology. In: Proceedings of the 8th Summer Conference on General Topology and Applications, vol. 728, pp. 183-197 (1994)

10. Nastasi, A, Vetro, P: Fixed point results on metric and partial metric spaces via simulation functions. J. Nonlinear Sci. Appl.8, 1059-1069 (2015)

11. Oltra, S, Valero, O: Banach’s ﬁxed point theorem for partial metric spaces. Rend. Ist. Mat. Univ. Trieste36, 17-26 (2004) 12. Paesano, D, Vetro, P: Suzuki’s type characterizations of completeness for partial metric spaces and ﬁxed points for

partially ordered metric spaces. Topol. Appl.159(3), 911-920 (2012)

13. Romaguera, S: Matkowski’s type theorems for generalized contractions on (ordered) partial metric spaces. Appl. Gen. Topol.12, 213-220 (2011)

14. Romaguera, S: Fixed point theorems for generalized contractions on partial metric spaces. Topol. Appl.159(1), 194-199 (2012)

15. Rus, IA: Fixed point theory in partial metric spaces. An. Univ. Vest. Timi¸s., Ser. Mat.-Inform..46(2), 141-160 (2008) 16. Samet, B, Vetro, C, Vetro, F: From metric spaces to partial metric spaces. Fixed Point Theory Appl.2013, 5 (2013) 17. Shahzad, N, Valero, O: On 0-complete partial metric spaces and quantitative ﬁxed point techniques in denotational

semantics. Abstr. Appl. Anal.2013, Article ID 985095 (2013)

18. Vetro, C, Vetro, F: Common ﬁxed points of mappings satisfying implicit relations in partial metric spaces. J. Nonlinear Sci. Appl.6, 152-161 (2013)

19. Vetro, C, Vetro, F: Metric or partial metric spaces endowed with a ﬁnite number of graphs: a tool to obtain ﬁxed point results. Topol. Appl.164, 125-137 (2014)